Finitely generated algebra

Short description: Type of algebra

In mathematics, a finitely generated algebra (also called an algebra of finite type) over a (commutative) ring $R$, or a finitely generated $R$-algebra for short, is a commutative associative algebra $A$ defined by ring homomorphism $f:R\to A$, such that every element of $A$ can be expressed as a polynomial in a finite number of generators $a_1,\dots ,a_n\in A$ with coefficients in $f(R)$. Put another way, there is a surjective $R$-algebra homomorphism from the polynomial ring $R[X_1,\dots ,X_n]$ to $A$.

If $K$ is a field, regarded as a subalgebra of $A$, and $f$ is the natural injection $K\hookrightarrow A$, then a $K$-algebra of finite type is a commutative associative algebra $A$ where there exists a finite set of elements $a_1,\dots ,a_n\in A$ such that every element of $A$ can be expressed as a polynomial in $a_1,\dots ,a_n$, with coefficients in $K$.

Equivalently, there exist elements $a_1,\dots ,a_n\in A$ such that the evaluation homomorphism at $\mathbf{𝐚}=(a_1,\dots ,a_n)$

${\varphi }_{\mathbf{𝐚}}:K[X_1,\dots ,X_n]\twoheadrightarrow A$

is surjective; thus, by applying the first isomorphism theorem, $A\cong K[X_1,\dots ,X_n]/\mathrm{k}\mathrm{e}\mathrm{r}({\varphi }_{\mathbf{𝐚}})$.

Conversely, $A:=K[X_1,\dots ,X_n]/I$ for any ideal $I\subseteq K[X_1,\dots ,X_n]$ is a $K$-algebra of finite type, indeed any element of $A$ is a polynomial in the cosets $a_i:=X_i+I,i=1,\dots ,n$ with coefficients in $K$. Therefore, we obtain the following characterisation of finitely generated $K$-algebras:

$A$ is a finitely generated $K$-algebra if and only if it is isomorphic as a $K$-algebra to a quotient ring of the type $K[X_1,\dots ,X_n]/I$ by an ideal $I\subseteq K[X_1,\dots ,X_n].$

Algebras that are not finitely generated are called infinitely generated.

A finitely generated ring refers to a ring that is finitely generated when it is regarded as a $\mathbb{\mathbb{Z}}$-algebra.

An algebra being finitely generated (of finite type) should not be confused with an algebra being finite (see below). A finite algebra over $R$ is a commutative associative algebra $A$ that is finitely generated as a module; that is, an $R$-algebra defined by ring homomorphism $f:R\to A$, such that every element of $A$ can be expressed as a linear combination of a finite number of generators $a_1,\dots ,a_n\in A$ with coefficients in $f(R)$. This is a stronger condition than $A$ being expressible as a polynomial in a finite set of generators in the case of the algebra being finitely generated.

Examples

• The polynomial algebra $K[x_1,\dots ,x_n]$ is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.
• The ring of real-coefficient polynomials $\mathbb{ℝ}[x]$ is finitely generated over $\mathbb{ℝ}$ but not over $\mathbb{\mathbb{Q}}$.
• The field $E=K(t)$ of rational functions in one variable over an infinite field $K$ is not a finitely generated algebra over $K$. On the other hand, $E$ is generated over $K$ by a single element, $t$, as a field.
• If $E/F$ is a finite field extension then it follows from the definitions that $E$ is a finitely generated algebra over $F$.
• Conversely, if $E/F$ is a field extension and $E$ is a finitely generated algebra over $F$ then the field extension is finite. This is called Zariski's lemma. See also integral extension.
• If $G$ is a finitely generated group then the group algebra $KG$ is a finitely generated algebra over $K$.
• "Quadratic and quaternion algebras are essentially all the associative R-algebras which are finitely generated projective R-modules and have a standard involution."

Properties

• A homomorphic image of a finitely generated algebra is itself finitely generated. However, a similar property for subalgebras does not hold in general.
• Hilbert's basis theorem: if $A$ is a finitely generated commutative algebra over a Noetherian ring then every ideal of A is finitely generated, or equivalently, $A$ is a Noetherian ring.

Relation with affine varieties

Finitely generated reduced commutative algebras are basic objects of consideration in modern algebraic geometry, where they correspond to affine algebraic varieties; for this reason, these algebras are also referred to as (commutative) affine algebras. More precisely, given an affine algebraic set $V\subseteq {\mathbb{𝔸}}^n$ we can associate a finitely generated $K$-algebra

$\mathrm{\Gamma }(V):=K[X_1,\dots ,X_n]/I(V)$

called the affine coordinate ring of $V$; moreover, if $\varphi :V\to W$ is a regular map between the affine algebraic sets $V\subseteq {\mathbb{𝔸}}^n$ and $W\subseteq {\mathbb{𝔸}}^m$, we can define a homomorphism of $K$-algebras

$\mathrm{\Gamma }(\varphi )\equiv {\varphi }^{*}:\mathrm{\Gamma }(W)\to \mathrm{\Gamma }(V),{\varphi }^{*}(f)=f\circ \varphi ,$

then, $\mathrm{\Gamma }$ is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated $K$-algebras: this functor turns out to be an equivalence of categories

$\mathrm{\Gamma }:(\text{affine algebraic sets}{)}^{\mathrm{o}\mathrm{p}\mathrm{p}}\to (\text{reduced finitely generated }K\text{-algebras}),$

and, restricting to affine varieties (i.e. irreducible affine algebraic sets),

$\mathrm{\Gamma }:(\text{affine algebraic varieties}{)}^{\mathrm{o}\mathrm{p}\mathrm{p}}\to (\text{integral finitely generated }K\text{-algebras}).$

Finite algebras vs algebras of finite type

We recall that a commutative $R$-algebra $A$ is a ring homomorphism $\varphi :R\to A$; the $R$-module structure of $A$ is defined by

$\lambda \cdot a:=\varphi (\lambda )a,\lambda \in R,a\in A.$

An $R$-algebra $A$ is called finite if it is finitely generated as an $R$-module, i.e. there is a surjective homomorphism of $R$-modules

$R^{{\oplus }_n}\twoheadrightarrow A.$

Again, there is a characterisation of finite algebras in terms of quotients:

An $R$-algebra $A$ is finite if and only if it is isomorphic to a quotient $R^{{\oplus }_n}/M$ by an $R$-submodule $M\subseteq R$.

By definition, a finite $R$-algebra is of finite type, but the converse is false: the polynomial ring $R[X]$ is of finite type but not finite. However, if an $R$-algebra is of finite type and integral, then it is finite. More precisely, $A$ is a finitely generated $R$-module if and only if $A$ is generated as an $R$-algebra by a finite number of elements integral over $R$.

Finite algebras and algebras of finite type are related to the notions of finite morphisms and morphisms of finite type.

See also

• Finitely generated module
• Finitely generated field extension
• Artin–Tate lemma
• Noether normalization lemma
• Finite algebra
• Morphism of finite type